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X-ray pattern from a single crystal with c axis parallel to the X-ray beam. You can see its 3-fold symmetry, perhaps evidence of 3m. Laue photograph named after the first scientist who, in 1912, showed that X- rays are diffracted by crystals.

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Bragg’s law: for constructive interfence to occur, the path difference (AB+CD) among waves scattered by a set of lattice planes must equal a whole number (n=1, 2,...) of wavelengths. Only some angles theta will give you this result.

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E: this tube is the X-ray source. Inside it, there is a 40,000 volt difference between a tungsten filament and a copper target.

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Early X-ray diffraction powder camera. Film is rolled around the inner rim and records myriad of diffracted beams as semi- circular sections of cones.

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X-ray spectra used to be recorded on film strips rolled up within a round chamber.

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The distance from the center of each line to the center of the hole (where X-rays entered the chamber) is proportional to the angle 2-theta. The intensities of the lines were originally estimated by a human eye, on a scale of 1 to 100, before electronic detectors became routine.

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The first information we get from XRD is whether or not the solid being investigated is crystalline or amorphous. Silica glass, for example, has SiO 4 tetrahedra like quartz. Synthetic (i.e. human-made) quartz is indistinguishable from natural quartz by XRD if their structure (space group) and composition are the same.

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The trick is to relate each diffracted peak to the right family of planes (hkl). The job gets easier in minerals with a high degree of symmetry, because there is only a relatively small number of possible interplanar spacings. d (100) = d (010) = d (001) So all these planes diffract at the same theta angle.

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Bragg’s law: n = 2 d sin  1) What do we know?, i.e. the wavelenth of the X-ray radiation 2) What do we assume? n = 1 (Peaks for higher “n” are weaker.) 3) What do we want to know? d, i.e. the interplanar spacings of the lattice

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(1 angstrom = 10 –8 cm)

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The unit cell is described as being the smallest regular repeat unit in a crystalline lattice. These cells are defined by three unit lengths (a, b, c) along the crystallographic axe,s and the three interaxial angles ( , ,  ).

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The patterns with higher symmetry (and more nodes per unit cell) produce a lesser number of diffracted peaks, even if the interplanar spacings (100), (010) and (001), which describe the unit cell, are unchanged.

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Indices of diffracted X-ray peaks are usually written without parentheses. 111, 222 and 333 correspond to the 1 st, 2 nd and 3 rd order reflections of the (111) planes. 222 is produced when the X-rays of successive planes have a path difference of 2*wavelength (two “lambdas”).

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In non-primitive cells there are additional lattice planes, usually half-way between the usual lattice planes. They are offset by translations, but they have the same atomic pattern, so they will diffract X-rays just like the other planes. However, the phase difference will lead to negative interference, i.e. they will be a half-wavelength behind the X-rays diffracted by other sets of planes.

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Top and bottom planes diffract “in phase” -the crests of waves are lined up -The middle plane diffracts out of phase. Its trough cancels the crest of the plane above. - negative interfence = no diffracted beam

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This cancellation of diffraction peaks is called a “systematic absence”. The path difference at the usual theta- angle value is exactly half of one wavelength. When waves are exactly “out of phase”, you get negative interference. Each lattice plane cancels the peak diffracted by the next lattice plane.

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The patterns with higher symmetry (and more nodes per unit cell) produce a lesser number of diffracted peaks, even if the interplanar spacings (100), (010) and (001), which describe the unit cell, are unchanged.

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Powder X-ray diffraction is a routine technique to measure the amount of crystalline SiO 2 (quartz) present in mineral dust or soil. A chemical analysis will not distinguish the SiO 2 of quartz from the silicate portion present in the structure of clays and many other minerals.

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Even when a single mineral is present, a chemical analysis may not tell you what that mineral is.... This Anglo-Saxon brooch contains an inlay of CaCO 3, but is it calcite or aragonite (2 common polymorphs)?

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Bragg’s law predicts at which angles the peaks will be diffracted, but not their intensities. Diffraction intensities are influenced by the atomic number (Z) of the atoms in the structure, by the shape and size of the specimen, and by other factors related to the machine. We use the peak intensities to determine where the atoms are in the unit cell.

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Because each mineral is different from all others in either its chemistry or the geometric pattern of its atomic arrangement (space group), each powder XRD pattern is a fingerprint. Often, the three most intense peaks and their theta-angle are all that is needed to fingerprint a mineral, even in a mixture. A database is searched by a fast computer to match known patterns to the peaks measured from an unknown.

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100, 200, 300 are n=1, 2, 3... in Bragg’s law But they all come from the (100) planes.

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Single-crystal work is used for specialized purposes. One is to determine the space group. You need to use all the information available to orient your crystal along the axes of symmetry. You then check how much symmetry is present. Unfortunately, XRD always adds a center of symmetry to the pattern.

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This four-circle diffractometer is used to mount a single crystal, and rotate it in space. A detector moves around it to measure the position (theta-angle) and intensity of diffracted peaks.

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How to solve crystal structures? The electron density ( ) at a point X, Y, Z in a unit cell of volume “V” is; (X,Y,Z) = 1/V F hkl cos [ 2 (h  X + k  Y + l  Z) - ] Therefore if we know Fhkl and (for each h, k, l) we can compute for all values of X, Y, and Z and plot the values obtained to give a three- dimensional electron density map. Then, assuming atoms to be at the centres of the electron density peaks, we would have the entire structure.

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The presence or absence of a center of inversion is usually determined from properties such as : -presence of polar forms (e.g. pyramids, monohedra) which indicate that the “+” end of a crystallographic axis is different from the “-” end. - piezoelectricity which can only exist in crystalline structures having at least one polar axis.

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Laue photographs are used to study the epitaxial relationships between thin films and the material on which they are grown. Large spots: aluminum. Small spots: silicon.

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When detecting twinning matters ! Piezoelectric crystals may not display that property if they are twinned. Twinning can show up in - external forms - re-entrant angles (non-convex morphology)

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Ion order-disorder can be detected by X-ray diffraction. This is very different from the lack of order found in an amorphous solid.

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A cathode filament is heated so that it boils off electrons. A large voltage ( kV) is maintained between the filament and the target (a metal such as Mo, Cu, Co, Fe or Cr). The electrons are accelerated and hit the target metal.

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Structures with lighter elements can be studied using neutron diffraction. Neutrons are scattered by the nucleus, and their scattering varies less from element to element. whereas X-rays are scattered by the electron cloud, and light elements barely re- emit them.